Larson–Miller correlation for the effect of thermal ageing on the yield

strength of a cold worked 15Cr–15Ni–Ti modified austenitic stainless steel

K.G. Samuel *, S.K. Ray

Materials Technology Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, India

Received 11 November 2005; received in revised form 16 February 2006; accepted 20 February 2006

Abstract

For 20% cold worked 15Cr–15Ni–Ti modified austenitic stainless steel (Alloy D9), the Larson–Miller parameter can be used to describe the

effects of prior thermal exposures to different time–temperature combinations on the 0.2% yield stress sYS, ultimate strength and total elongation

in subsequent tensile tests at 300, 723 and 923 K. A single master plot for all the tensile test temperatures was obtained by plotting the Larson–

Miller parameter against the ratio SYSZ(sYS of thermally aged material)/(sYS of un-aged material) at identical tensile testing temperature.

q 2006 Elsevier Ltd. All rights reserved.

Keywords: Ti modified austenitic stainless steel; Cold work; Thermal ageing; Larson–Miller parameter

1. Introduction

The Larson–Miller parameter, PZT(log10 tCC), where T is

the absolute temperature, t the time and C a constant, had its

origin in the tempering studies of Hollomon and Jaffe [1]. This

parameter continues to be widely used for correlation of stress

rupture data of various engineering materials [2,3]. The

Larson–Miller parametric correlation has also been used for

hardness and notch toughness of 2.25Cr–1Mo steel [4], the

influence of ageing on the hardness of cold-worked austenitic

stainless steel [5] and carbon concentration profiles in Alloy

800H/2.25Cr–1Mo steel joints welded with Inconel 82

consumables [6].

Titanium modified 15Cr–15Ni austenitic stainless steel

(Alloy D9) is chosen for the hexagonal wrapper for fuel

subassemblies of fast breeder reactors [7]. This material is

generally used in a 20% prior cold worked condition, and there

is an interest in assessing the influence of elevated temperature

service exposure on the tensile deformation behaviour,

specifically the 0.2% yield stress sYS, ultimate strength and

total elongation. Vasudevan et al. [8] have extensively studied

the recovery and recrystallization behaviour on static thermal

ageing 20% cold worked 15Cr–15Ni–2.2Mo–Ti modified

austenitic steel with various Ti/C ratios, using optical

metallography, and room temperature hardness measurements

and tension tests. They showed that recrystallization during

prior static ageing leads to drastic decreases in hardness and

strength values with corresponding increase in the elongation.

The recrystallization temperature was found to be w973 K,

considerably accelerated as ageing temperature increased, and

depended on the Ti/C ratio. Metallographic observations [8]

indicated the presence of grain boundary precipitates in the

thermally aged alloys. In cold worked and thermally aged steel

of this type, grain boundary precipitates of the type M23C6 and

MC have been reported [9,10]. In this paper, it is shown that the

Larson–Miller parameter can be used to describe the effects of

static thermal exposure of 20% cold worked Alloy D9 on the

subsequent tensile properties at 300, 723 and 923 K.

2. Experimental

The dimensions of the hexagonal wrapper tube are

131.3 mm wide across flat faces and 3.2 mm thickness. The

chemical composition (wt%) of the material investigated was

C: 0.045, Cr: 13.88, Ni: 15.24, Mo: 2.12, Ti: 0.23, B: 12 ppm,

Mn: 2.12, Si: 0.64, Cu: 0.017, As: 0.0019, N: 0.0021, Al: 0.01,

Co: 0.007, S:!0.005, P:!0.005, Nb:!0.005, V:!0.01,

Ta:!0.01. The tubes were procured in the 20G4% cold

worked condition. Tensile specimen blanks were cut from the

flat faces of the wrapper tube in the axial direction and given an

isothermal ageing treatment at a temperature in the range

823–1123 K for various durations up to 10,000 h, and then

quenched in water to retain the microstructure developed

International Journal of Pressure Vessels and Piping 83 (2006) 405–408

www.elsevier.com/locate/ijpvp

0308-0161/$ - see front matter q 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijpvp.2006.02.032

* Corresponding author.

E-mail address: samuel@igcar.gov.in (K.G. Samuel).

during ageing. In all, 32 ageing conditions (Table 1) were used

in this study. Flat tensile specimens having 25 mm gauge

length and 4 mm gauge width were machined from the unaged

as well as aged blanks.

Isothermal tensile tests were carried out in a universal

testing machine at a constant cross head speed of 2 mm/min

(nominal strain rateZ1.33!10K3 sK1). The load and elonga-

tion were recorded using the chart drive attached

to the machine. The elevated temperature was controlled

within G2 K over the gauge length using a three zone

resistance-heating furnace. Prior cold worked (PCW) and prior

cold worked and aged (PCWA) specimens were tested at 300,

723 and 923 K.

3. Results and discussion

The typical true stress strain curves for the aged materials

are compared with that of the as received material in Fig. 1. It is

observed that particularly higher ageing temperatures and

longer durations lead to substantial changes in strength and

ductility, reflecting changes in microstructure during static

thermal ageing. The average values (from a minimum of two

tests) of sYS of the cold worked material as a function of ageing

conditions and test temperature are shown in Table 1. The

combined effects of thermal ageing is sought to be expressed

using the Larson–Miller parameter

PZ TAðlog10 ta CCÞ (1)

where TA (in K) and ta (in h), are, respectively, ageing

temperature and duration and C, a constant to be determined.

Using this parameter, the dependence of sYS (in MPa) on prior

thermal ageing could be expressed using a polynomial of

degree 3:

sYS Z a0 Ca1PCa2P2Ca3P

3 (2)

The degree of the polynomial was fixed as optimal by trial

and error. For a fixed value of C, the polynomial coefficients in

Eq. (2) were determined from a least squares fit. The value of

the constant C was varied to identify the polynomial fit that

gave the highest correlation coefficient R. The variations of the

correlation coefficient R with the Larson–Miller parameter

constant C are shown in Fig. 2. The C value corresponding to

the highest correlation coefficient was found to be 13,

independent of the tensile test temperature. The optimal values

for the constants in Eq. (2) thus determined are shown in

Table 2.

The variation of sYS with P for the three tensile test

temperatures is shown in Fig. 3; the firm lines in this figure

represent the optimal fits to Eq. (2). As this figure shows, a

separate correlation is obtained for each of the tensile test

temperatures, although with very similar trends in variation of

sYS with P: initially a slight increase, followed by a rapid

decrease, followed by a trend to saturation at large P values.

Table 1

Yield strength sYSa of 20% prior cold worked Alloy D9 after thermal ageing

Ageing

time (h)

Aged at 823 K Aged at 923 K Aged at 1023 K Aged at 1123 K

300 K 723 K 923 K 300 K 723 K 923 K 300 K 723 K 923 K 300 K 723 K 923 K

10 651 623 520 646 553 517 614 473 445 508 402 384

50 707 563 525 720 574 510 588 494 425 399 323 307

100 666 588 491 727 560 545 565 443 416 390 235 197

500 685 543 474 663 517 473 471 368 357 257 200 167

1000 722 547 473 614 473 445 524 401 376 236 163 141

2000 718 562 485 588 494 425 429 323 302 237 161 142

5000 659 529 469 534 422 369 353 290 277 231 179 149

10,000 648 496 469 501 397 370 321 267 235 224 159 142

asYS values in MPa.

8 10 12 14 16 18 20 22 24 260.90

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98TESTTEMPERATURE

923 K

723 K

300 K

CO

RR

ELA

TIO

N C

OE

FF

ICIE

NT

, R

L-M PARAMETER CONSTANT, C

Fig. 2. Variation of correlation coefficient for fit with Eq. (1) as a function of

Larson–Miller parameter constant C at various test temperatures.

0.00 0.05 0.10 0.15 0.20 0.25100

200

300

400

500

600

700

800

900

1000

TestTemperature

Ageing ConditionTemperature/Time

A 300 K 20% Prior Cold Worked (PCW)B 300 K PCW + 823 K/10 hoursC 923 K 20% Prior Cold Worked (PCW)D 923 K PCW + 1123 K/2000 hours

D

C

B

A

TR

UE

ST

RE

SS

, MP

a

TRUE PLASTIC STRAIN

Fig. 1. Typical stress–strain curves of Ti modified austenitic stainless steel in

cold worked and subsequent thermal ageing conditions at 300 and 923 K.

K.G. Samuel, S.K. Ray / International Journal of Pressure Vessels and Piping 83 (2006) 405–408406

A similar trend has been observed by Vasudevan et al. [5] in

their study on the dependence of hardness of prior cold worked

D9 alloy subjected to prior thermal ageing using the Larson–

Miller parametric approach. The slight initial increase in sYS

with P might be attributed to precipitation of TiC during

ageing. Kesternich and Meertens [11] have reported that fine

homogenous dispersions of TiC were obtained in a 20% cold

worked Ti stabilized 15Cr–15Ni austenitic stainless steel on

ageing at 923 K. The decrease in sYS at higher P values reflects

the recovery and possibly recrystallization of the initial cold

worked structure during thermal ageing.

sYS depends both upon the microstructure developed during

the prior thermal ageing, and also the temperature (and strain

rate) for the tensile test. However, it may be possible to index

the microstructure variation using a structure sensitive yield

strength ratio SYS defined as

SYS Z ðsYS for aged materialÞ=ðsYS for unaged materialÞ (3)

at an identical tensile test temperature (and strain rate). For a

Type 316 stainless steel Samuel et al. [12] showed the viability

of this yield strength ratio, and also the ratio of tensile ductility

defined in an analogous manner, for indexing the effects of

prior cold work in various modes of deformation.

Fig. 4 shows the plots of SYSwith P, which follows the same

three-stage pattern observed for each tensile test temperature in

Fig. 3. The optimal correlation between SYS and P was

determined in the manner described above, as:

SYSZK14:636C0:0031PK2:615!10K7P2C4:096!10K12

P3

(4)

Here too, the optimal value for C was determined to be 13.

The possibility of generating master curves such as Eq. (4)

for ultimate tensile strength ratio, SUTS and total elongation

ratio, STE defined in analogous ways to Eq. (3) was explored.

The results for these analyses are shown in Figs. 5 and 6,

respectively. Clearly, TA and ta can be combined using the

Larson–Miller parametric approach to describe the effects of

prior thermal ageing on ultimate tensile strength sUTS and total

elongation TE. The optimal values for the corresponding

10000 11000 12000 13000 14000 15000 16000 17000 18000 19000 20000100

200

300

400

500

600

700

800

YIE

LD S

TR

EN

GT

H, M

Pa

P = TA [log10 (ta/h) + 13]

Tension Test Temperature

300 K723 K923 K

Fig. 3. Variation of yield strength with Larson–Miller parameter at various test

temperatures.

11000 12000 13000 14000 15000 16000 17000 18000 19000 200000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2 Tension Test Temperature

300 K 723 K 923 K

YIE

LD S

TR

EN

GT

H R

AT

IO, S

YS

P = TA [Log(ta/h) + 13]

Fig. 4. Variation of yield strength ratio with Larson–Miller parameter at various

test temperatures.

10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 300000.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2Test Temperature C

300 K 18 723 K 22 923 K 12

TE

NS

ILE

DU

CT

ILIT

Y R

AT

IO, S

TE

P = TA [Log(ta/h) + C]

Fig. 6. Variation of tensile ductility ratio with Larson–Miller parameter at

various test temperatures.

10000 12000 14000 16000 18000 20000 22000 24000 26000

0.550.600.650.700.750.800.850.900.951.001.051.101.151.20

Test Temperature C 300 K 18 723 K 12 923 K 12

TE

NS

ILE

ST

RE

NG

TH

RA

TIO

, SU

TS

P = TA [Log(ta/h) + C]

Fig. 5. Variation of tensile strength ratio with Larson–Miller parameter at

various test temperatures.

Table 2

Polynomial constants in Eq. (2) for the fits to sYS data in Fig. 3

Test

temperature

(K)

Polynomial constants

a0 a1 a2 a3

300 K13160.076 2.749 K1.765!10K4 3.622!10K9

723 K5796.868 1.303 K8.494!10K5 1.778!10K9

923 K6711.446 1.434 K9.133!10K5 1.830!10K9

K.G. Samuel, S.K. Ray / International Journal of Pressure Vessels and Piping 83 (2006) 405–408 407

constants are given in Tables 3 and 4. However, master plots

using SUTS or STE could not be obtained. It may be noted that

the result for STE (Fig. 6) is in contrast to that of Samuel et al.

[12] for cold worked SS 316. It is, however, interesting to note

that the SUTS data for 723 and 923 K apparently can be

considered to belong to a single band (Fig. 5), while STE data

for 300 and 723 K seem to belong to a single band (Fig. 6).

These trends could actually be anticipated from the

corresponding optimal values for C, Tables 3 and 4 and

Figs. 5 and 6. The mechanistic reasons for these observations

are not clear. sYS depends upon the initial microstructure and

therefore the parameter SYS is successful in indexing the

microstructure developed after the static ageing. Tensile

deformation apparently results in significant modulation

of deformation and damage substructure. These modulations

determine sUTS and TE, but are not reflected in P, and

therefore SUTS and STE cannot be used to index the

microstructure. Differences in the observed dependences of

SUTSKP and STEKP relations (Figs. 5 and 6) however suggest

that a single parametric formulation would not succeed for both

SUTS and STE. More detailed modelling is needed to sort out

this issue.

4. Conclusions

1. Variation of the 0.2% yield strength of a 20% prior cold

worked Alloy D9 at 300–923 K after static thermal

ageing for ta hours at temperature of TA (ta%104 h,

823%TA%1123 K) could be adequately described by the

Larson–Miller parameter, PZTA(log10 taCC), with the

optimal value of C as 13.

2. For tensile tests at 300, 723 and 923 K, a master curve,

independent of tensile test temperature, is obtained when

the ratio of the yield strengths of the aged and unaged

materials at identical test temperature is plotted against the

Larson–Miller parameter defined. It is concluded that this

ratio quantitatively indexes the microstructure at the start of

the tensile test. For this correlation too, the optimal value of

C is 13.

3. The Larson–Miller parameter correlation adequately

described the effect of prior thermal ageing on both

ultimate strength and total elongation in the subsequent

tensile tests at 300, 723 and 923 K. However, correspond-

ing ratios failed to yield master curves independent of

tensile test temperature.

References

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Trans AIME 1945;162:223.

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[4] Wignarajah S, Masumoto I, Hara. Evaluation and simulation of the

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[5] Vasudevan M, Venkadesan S, Sivaprasad PV, Mannan SL. Use of

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Table 4

Polynomial constants in Eq. (2) for the fits to TE data

Test

temperature

(K)

Polynomial constants

a0 a1 a2 a3

300 595.962 K0.074 3.032!10K6K3.612!10K11

723 K271.965 0.042 K2.081!10K6 3.425!10K11

923 5.898 0.006 K8.498!10K7 3.668!10K11

Table 3

Polynomial constants in Eq. (2) for the fits to sUTS data

Test

temperature

(K)

Polynomial constants

a0 a1 a2 a3

300 K7805.158 1.268 K6.065!10K5 9.308!10K10

723 K1136.072 0.437 K3.307!10K5 7.476!10K10

923 K5534.991 1.342 K9.602!10K5 2.199!10K9

K.G. Samuel, S.K. Ray / International Journal of Pressure Vessels and Piping 83 (2006) 405–408408